SU(N) caloron measure and its relation to instantons

TL;DR

This paper derives the measure for SU(N) calorons with non-trivial holonomy, relating it to instantons and their constituent monopoles, and shows consistency with standard instanton measures in certain limits.

Contribution

It provides an explicit computation of the caloron measure in terms of monopole constituents and connects it to the traditional instanton measure in specific regimes.

Findings

Calorons decompose into N BPS monopoles with non-trivial holonomy.

The caloron measure reduces to the standard instanton measure in the limit of small dyon separation.

Explicit relations between instanton and dyon collective coordinates are established.

Abstract

Calorons of the SU(N) gauge group with non-trivial holonomy, i.e. periodic instantons with arbitrary eigenvalues of the Polyakov line at spatial infinity, can be viewed as composed of N Bogomolnyi--Prasad--Sommerfeld (BPS) monopoles or dyons. Using the metric of the caloron moduli space found previously we compute the integration measure over caloron collective coordinates in terms of the constituent monopole positions and their U(1) phases. In the limit of small separations between dyons and/or trivial holonomy, calorons reduce locally to the standard instantons whose traditional collective coordinates are the instanton center, size and orientation in the color space. We show that in this limit the instanton collective coordinates can be explicitly written through dyons positions and phases, and that the N-dyon measure coincides exactly with the standard instanton one.